Yes, it is cylinder by cylinder.
All the surfaces belong to each cylinder. For example, in the figure shown above, consider that there are four surfaces (on each side of platter). So, every cylinder has a (part of) each surface.
If the hard disk is partitioned, then from say cylinder 0 to cylinder 1000 may belong to one partition while from cylinder 1001 to cylinder 4000 may belong to another. So different cylinders can be filled without the need to wait for one cylinder to finish.